Counting perfect matchings in n-extendable graphs

The structural theory of matchings is used to establish lower bounds on the number of perfect matchings in n-extendable graphs. It is shown that any such graph on p vertices and q edges contains at least ⌈(n+1)!/4[q-p-(n-1)(2Δ-3)+4]⌉ different perfect matchings, where Δ is the maximum degree of a vertex in G.     

Finding more perfect matchings in leapfrog fullerenes

We use some recent results on the existence of long cycles in leapfrog fullerenes to establish new exponential lower bounds on the number of perfect matchings in such graphs. The new bounds are expressed in terms of Fibonacci numbers.     

Decoherence in quantum mechanics

We study decoherence in a simple quantum mechanical model using two approaches. Firstly, we follow the conventional approach to decoherence where one is interested in solving the reduced density matrix from the perturbative master equation. Secondly, we consider our novel correlator approach to decoherence where entropy is generated by neglecting observationally inaccessible correlators. We show that both methods can accurately predict decoherence time scales. However, the perturbative master equation generically suffers from instabilities which prevents us to reliably calculate the system’s total entropy increase. We also discuss the relevance of the results in our quantum mechanical model for interacting field theories.     

Resolving Curvature Singularities in Holomorphic Gravity

We formulate a holomorphic theory of gravity and study how the holomorphy symmetry alters the two most important singular solutions of general relativity: black holes and cosmology. We show that typical observers (freely) falling into a holomorphic black hole do not encounter a curvature singularity. Likewise, typical observers do not experience Big Bang singularity. Unlike Hermitian gravity (Mantz and Prokopec in , 2008), holomorphic gravity does not respect the reciprocity symmetry and thus it is mainly a toy model for a gravity theory formulated on complex space-times. Yet it is a model that deserves a closer investigation since in many aspects it resembles Hermitian gravity and yet calculations are simpler. Our study of light bending and gravitational waves in weak holomorphic gravitational fields strongly suggests that holomorphic gravity reduces to general relativity at large distance scales.     

Exact treatment of generalized modifications of finite-dimensional systems by the LRM approach

LRM (Low Rank Modification) is a mathematical method that produces eigenvalues and eigenstates of generalized eigenvalue equations. It is similar to the perturbation expansion in that it assumes the knowledge of the eigenvalues and eigenstates of some related (unperturbed) system. However, unlike perturbation expansion, LRM produces correct results however large the modification of the original system. LRM of finite-dimensional systems is here generalized to the combined (external and internal) modifications. Parent n-dimensional system A _n containing n eigenvalues λ _i and n eigenstates \({| {\Phi_i}\rangle}\) is described by the generalized n × n eigenvalue equation. In an external modification system A _n interacts with another ρ-dimensional system B _ρ which is situated outside the system A _n . In an internal modification relatively small σ-dimensional subsystem of the parent system A _n is modified. Modified system C _n+ρ that contains external as well as internal modifications is described by the generalized (n + ρ) × (n + ρ) eigenvalue equation. This system has (n + ρ) eigenvalues \({\varepsilon_s}\) and (n + ρ) corresponding eigenstates \({| {\Psi_s}\rangle}\) . In LRM this generalized (ρ + n) × (ρ + n) eigenvalue equation is replaced with a (nonlinear) (ρ + σ) × (ρ + σ) equation which produces all eigenvalues \({\varepsilon_s \notin \left\{ {\lambda_i}\right\}}\) and all the corresponding eigenstates \({| {\Psi_s}\rangle }\) of C _{n + ρ}. Another equation produces remaining solutions (if any) that satisfy \({\varepsilon_s \in \left\{ {\lambda_i}\right\}}\) . Those two equations produce exact solution of the modified system C _{n + ρ}. If (ρ + σ) is small with respect to n, this approach is numerically much more efficient than a standard diagonalization of the original generalized eigenvalue equation. Unlike perturbation expansion, LRM produces exact results, however large modification of the parent system A _n .     

Rhodium(III)-catalyzed oxidative carbonylation of benzamides with carbon monoxide

An efficient strategy for the oxidative carbonylation of aromatic amides via C-H/N-H activation to form phthalimides using an Rh(III) catalyst has been developed. The reaction shows a preference for C-H bonds of electron-rich aromatic amides and tolerates a variety of functional groups.     

A modular approach to the synthesis of 2,3,4-trisubstituted tetrahydrofurans

A highly diastereoselective Lewis acid-mediated [1,3] rearrangement of 1,3-dioxepins is the key step along a modular route to 2,3,4-trisubstituted tetrahydrofurans.     

The newtonian limit of hermitian gravity

We construct the gauge invariant potentials of Hermitian Gravity [1] and derive the linearized equations of motion they obey. A comparison reveals a striking similarity to the Bardeen potentials of general relativity. We then consider the response to a point particle source, and discuss in what sense the solutions of Hermitian Gravity reduce to the Newtonian potentials. In a rather intriguing way, the Hermitian Gravity solutions exhibit a generalized reciprocity symmetry originally proposed by Born in the 1930s. Finally, we consider the trajectories of massive and massless particles under the influence of a potential. The theory correctly reproduces the Newtonian limit in three dimensions and the nonrelativistic acceleration equation. However, it differs from the light deflection calculated in linearized general relativity by 25 %. While the specific complexification of general relativity by extension to Hermitian spaces performed here does not agree with experiment, it does possess useful properties for quantization and is well-behaved around singularities as described in [1]. Another form of complex general relativity may very well agree with experimental data.